Аннотация:
Let $R$ be a regular semi-local ring containing an infinite perfect subfield and let $K$ be its field of fractions. Let $G$ be a reductive $R$-group scheme satifying a mild “isotropy condition”. Then each principal $G$-bundle $P$ which becomes trivial over $K$ is trivial itself. If $R$ is of geometric type, then it suffices to assume that $R$ is of geometric type over an infinite field. Two main Theorems of Panin, Stavrova and Vavilov proven recently state the same results for semi-simple simply connected $R$-group schemes. Our proof is heavily based on those two theorems and on a classical result of Colliot–Théléne and Sansuc.
